So what are some real-world benefits of complexity theory?
Macro-economic theories are based on high-level relationships that economists have observed in reality, and that have some theoretical basis. For example, supply and demand can be represented as curves (aggregates of individual positions), and that in a perfect market the value of a good will converge at the intercept of supply and demand. There are a considerable number of assumptions required to make a statement like this, most of which are made to 'manage away' complexity.
Complexity theory helps us find ways to address and model complexity, such that we can reduce the number of assumptions required. For example, if we can build an agent-based simulation of supply and demand, and we model these agents as realistically as possible (few assumptions), we can then see if the same realisation of a good's value occurs.
The value in this is that we can then test the assumptions directly (what happens if 20% of people make irrational decisions?) and see how it affects the emergent behaviour.
Complexity theory can help in properly building these kinds of fine-grained simulations in a smart way, and can also help in understanding and debugging such simulations when the wrong behaviour emerges.
In my field (power systems) complexity theory can help us investigate and understand why large-scale blackouts have a fat tail distribution. That is, to understand why our macro-level statistical models aren't accurate in the edge cases. Primarily it is because these outliers occur 'emergently' due to the complexity of power systems, in ways that we overlook by applying old-school approaches to grid development and system operation.
In short: complexity theory helps us look beyond our 'mostly right' high-level statistical models and theories.
The same intuitions could be applied to other things. Large scale power grids are also often hard to predict when not moving into a sure fail state. Being able to analyze how you stabilize these systems without basically dumping a lot of money on them is the way to go (Looking in the past, the money will probably not be spend).
You could study the behavior of crowds and maybe make estimations on the safety large conventions build a "panic index" that calculates the risk of having something like at the Loveparade https://en.wikipedia.org/wiki/Love_Parade_disaster . Again - you would not be able to make a precise prediction of whats gonna happen but I'd say it'd even knowing if you have like a 0.5% Chance of a disaster would be worth knowing. (Of course, there are effective methods taught to prevent this disasters anyway. But sometimes you have new configurations that didn't occur in the past and you might catch these things with a simulation. I could be an additional approach)
But a quick search brought this paper to the light which might serve as a starting point: https://res.mdpi.com/d_attachment/energies/energies-11-01381...
For example they report on a project where they used photovoltaic panels to stabilize changing power consumption in a power grids with minimal changes in the existing structure - something that is hard with traditional power plants since they basically have to much momentum for quick switching action. https://ieeexplore.ieee.org/document/7007647
Also, agent based simulations (building on the idea of self organized criticality) are a thing. With these setups you can test power grids on their reaction for certain failure types - something you don't want to test in real life. I assume these simulations would also be quite accurate since consumption and production should be well known, as well as the physical properties of transmission.
So, the academically interesting aspect aside, it can spur research for useful things, it can act as a guide to what is worth / not worth researching (if something is proven to be in a "hard" complexity class, then you won't find any efficient algorithm to solve the problem, ever (assuming P =/= NP).) and is generally a great framework for algorithms research. And evidently, algorithms create a lot of value (good and bad, unfortunately).
It's just very much "under the hood". 100 years ago people could have also, rightly, said "how does quantum theory benefit people, what's the value in that". Yet, without the fundamental research in quantum theory we would not have modern technology. Understanding the behaviour of electrons in materials on an atomic level requires quantum mechanics.
In it are the beginnings of a new simple universal way to measure complexity.
The "real-world benefits" are that you can compare 2 systems that accomplish the same problem and objectively choose the less complex one, much like you could use the measurement of "weight" to pick a lighter material for something like a plane.
To give examples requires an agreement on exactly which domains of knowledge you consider to be part of "complex systems", so based on your question I'll assume that you consider chaos to be part of it.
For example, the Poincaré-Bendixson theorem saves researchers time because they don't have to consider the possibility of chaos in 2D continuous state space models.
For example, since we know that weather is a chaotic dynamical system, we can give up on the hope of ever being able to precisely predict the weather a year beforehand. This saves researchers from spending their time on impossible objectives.
And most things in business are neither truly linear nor simple exponential. Yet most people in business revert to these two mathematical concepts.
For example, it might convince a CEO to keep working on a project because this 'emergence' thing may be happening -- where a non-complexity-empowered CEO might kill the project because there is no linear return, and no apparent potential for exponential return.